TCHNICAL PAPR Journal of the South African Institution of Civil ngineering ISSN 11-19 Vol 59 No 1, March 17, Pages 8 34, Paper 1385 PROF JAMS MAINA (Pr ng, MSAIC, FSAA) is a professional pavement engineer, full-time professor of civil engineering at the Universit of Pretoria, and currentl on secondment as a technical director of a QA/QC Project for roads in the State of Qatar. He obtained his PhD from Miaaki Universit in Japan. His professional activities include QA/QC in road projects, pavement materials, and the development of advanced numerical analsis (modelling) tools for pavement engineering application. He also teaches both under- and post-graduate classes at the Universit of Pretoria. Contact details: Department of Civil ngineering Universit of Pretoria Private Bag X Hatfield 8 South Africa T: +7 1 4 668 : james.maina@up.ac.a PROF FUTOSHI KAWANA is an associate professor at the Toko Universit of Agriculture in Japan. His specialises in structural mechanics. He received his doctorate in engineering from the Science Universit of Toko in 4. He is a member of the Japan Societ of Civil ngineers, the Pavement Diagnosis Researchers Group (NPO) and the Japanese Societ of Irrigation, Drainage and Rural ngineering. Contact details: Department of Bioproduction and nvironment ngineering Facult of Regional nvironment Science Toko Universit of Agriculture 1-1-1 Sakuragaoka Setagaa-ku Toko, 156-855 Japan T: +81 3 5477 34 : fk56@nodai.ac.jp PROF KUNIHITO MATSUI is emeritus professor of civil and environmental engineering at Toko Denki Universit in Japan. He obtained his PhD from the Department of Mechanics and Hdraulics at the Universit of Iowa, USA, in 1974. His areas of interest include structural analsis in pavement structures, static and dnamic back-calculation of pavement sstems, non-destructive testing, thermal analsis of pavement sstems, parameter identification, sensitivit analsis and structural optimisation. Contact details: Department of Civil and nvironmental ngineering Toko Denki Universit Hatoama Hiki Satama, 35-394 Japan T: +81 49 96 573/549 : matsui@g.dendai.ac.jp Kewords: pavement, linear-elastic analsis, transversel isotropic, cross anisotrop, isotropic, rectangular loading Numerical modelling of fleible pavement incorporating cross anisotropic material properties Part II: Surface rectangular loading J W Maina, F Kawana, K Matsui In order to better understand the impact of increased loading on roads, studies on tre-road interaction have gained prominence in recent ears. Tres form an essential interface between vehicles and road pavement surfaces. These are the onl parts of the vehicle that are in contact with the road and transmit the vehicle loading to the road surface. The use of the Cartesian coordinate sstem is convenient in dealing with a uniform/non-uniform tre load acting over a rectangular area, but few research reports are available that provide an form of theoretical solutions for pavement responses. This paper presents analtical solutions of responses due to rectangular loading acting on the surface of a multi-laered pavement sstem. The solutions developed incorporate both isotropic and cross-anisotropic material properties. The method followed is based on classical trigonometric integral and Fourier transformation of Navier s equations. Accurac and validit of the solutions are verified through comparisons with a proprietar finite element method (FM) package. For this purpose, a pavement structure composed of five main laers constituted b isotropic and cross-anisotropic (also known as transversel isotropic) material properties is analsed. In order to var some of the laer properties with depth, the main laers were sub-laered, resulting in a 17-laer pavement sstem. INTRODUCTION Motivation for this work is discussed in the companion paper b Maina et al (17), (see pages 7 in this edition). Although modern era trucks transport heavier cargo, the are using relativel fewer tres than their predecessors, and as a result the are purported to be eerting much higher contact stresses on the road surface. A good understanding of tre-road contact stresses, and the abilit to model the macroscopic behaviour of materials when subjected to varing traffic loading and environmental conditions, is therefore important for better road pavement designs and improved pavement performance. Tres are the onl part of the vehicle that are in contact with the road, and transmit the vehicle loading to the road surface through a ver small contact area, called the contact patch or tre footprint. Generall, there are two main tpes of truck tres widel used on our roads the single (or so-called wide-base tre) and the conventional dual-truck tres. A single wide-base tre is a proportionatel larger and more robust tre that is now being used on trucks for heav cargo. This tpe of tre is epected to replace dual-tres in the future, on condition of minimal damage to the eisting road infrastructure. To be able to carr the same load as the dual-tres, the wide-base tre ma have a much greater tre inflation pressure and a larger individual footprint (but could also be smaller than the two combined footprints from standard dual-tres). Research done b De Beer (8) on tre-pavement contact stresses has also shown tre-pavement contact stresses to be, although dependent on the loading magnitude and inflation pressure, mostl rectangular and occasionall circular in shape. Development of solutions for circular surface loading has alread been presented in the companion paper published in this edition (Maina et al 17). In order to develop closed-form solutions for resilient responses of a pavement structure under the rectangular tre loading, the Cartesian coordinate sstem ma be convenient to use. Bufler (1971) derived the theoretical solution for multi-laered sstems using the Cartesian coordinate sstem for isotropic materials, but did not provide an worked eamples. Similarl, rnian (1989) used both the clindrical and Cartesian coordinate sstems to derive solutions for both circular and rectangular uniforml distributed loads acting on the surface of a multi-laered sstem with isotropic material properties. 8 Maina JW, Kawana F, Matsui K. Numerical modelling of fleible pavement incorporating cross-anisotropic material properties Part II: Surface rectangular loading. J. S. Afr. Inst. Civ. ng. 17:59(1), Art. #1385, 7 pages. http://d.doi.org/1.17159/39-8775/17/v59n1a4
p (,, ) p (,, ) p (,, ) (a) Vertical p (,, ) (b) Horiontal p (,, ) (c) Horiontal p (,, ) Figure 1 Uniforml distributed rectangular loads This paper presents the development of closed-form solutions for a multi-laered pavement sstem under static rectangular loadings, considering both isotropic and cross-anisotropic material properties. Isotropic materials have the same elastic properties in both the vertical and horiontal directions, and can be described b three independent elastic constants elastic modulus, Poisson s ratio v and shear modulus G. In contrast, cross-anisotrop of an elastic material is defined b five independent elastic constants two elastic moduli in vertical and horiontal directions ( v, h ), two Poisson s ratios in vertical and horiontal directions (v vh, υ hh ) and one shear modulus (G hv ), as presented b Love (1944). In this research stud, two classical mathematical methods, i.e. classical trigonometric integral and classical potential function, were investigated for fleibilit and efficienc. The former was adopted in this stud and its use in the determination of pavement responses is presented in this paper. This work is an etension of work where solutions of responses due to circular loading were presented (Maina et al 17). This method is fleible enough and can easil be etended to dnamic and wave propagation problems, although this is not the focus of the paper. Accurac and validit of the solutions are verified through comparisons of the computed responses to results obtained using a proprietar finite element package for a five-laer pavement structure, where the four upper main laers were sub-laered, resulting in a pavement sstem with 17 laers. The pavement laers were composed of materials with isotropic and cross-anisotropic (also known as transversel isotropic) properties. THORTICAL DVLOPMNT Three-dimensional linear elasticit For 3D problems shown in Figure 1, the differential equilibrium equations in Cartesian coordinates ma be epressed using modified Navier s equations. However, the equations are cumbersome to deal with because of the need to solve three coupled partial differential equations for the three displacement components. The difficult with finding particular solutions of the sstem of equations in terms of the displacements arises because each of the sought-after deflection functions in the Cartesian coordinates (, and ) appear in all three equilibrium equations. The solutions ma be simplified b representing displacements in terms of harmonic potentials. It is because this approach decouples the equations in various different was. The most common approach is to use the so-called Papkovich-Neuber potentials to represent the solution (Oawa et al 9; Borodachev & Astanin 8). This approach enables the use of a well-known catalogue of particular solutions of the Laplace equation, and sometimes even reduces the problem, if not completel, to one of the classical problems of the theor of harmonic functions (theor of potential). Despite the simplification, it is difficult to etend this approach to problems of dnamic or moving load analsis (Oawa et al 1). This paper aims at presenting closedform solutions of pavement responses due to static rectangular loading in the vertical direction. The solutions presented in this paper were derived based on a more fleible and efficient classical transform integral method. A similar approach can be followed to derive solutions for rectangular loads acting in the longitudinal and transverse directions. Theoretical development Three different approaches ma be used to solve problems of the theor of elasticit (Borodachev 1995; 1). In the first approach, the displacement vector is determined first, and this vector is then used to determine the stress and strain tensors (known as problem in displacements). Y In the second approach the stress tensor is determined first, and then this tensor is used to determine the strain tensor and displacement vector (known as problem in stresses). In the third approach the strain tensor is determined first, and then stress and displacement tensors are determined (known as problem in strains). The work presented in this paper followed the first approach, namel problem in displacements. A sstem of rectangular Cartesian coordinates (,, ) is used. B assuming the bod forces to be ero, equilibrium equations for an infinitesimal element (Figure ) can be epressed using Navier s equations as follows (Filonenko-Borodich 1963): σ + τ + τ = (1) τ τ + σ + τ Z Figure Stress distributions in a Cartesian coordinate sstem + τ + σ σ = () = (3) σ, σ, σ are normal stress and τ, τ and τ are shear stresses acting on an infinitesimal element. The two subscripts on the smbols for shear stresses represent, respectivel, the face and direction on which the shear stress is acting. X σ τ τ τ τ σ Journal of the South African Institution of Civil ngineering Volume 59 Number 1 March 17 9
The strain-displacement relationship ma be represented as follows: ε = u, ε = v, ε = w, γ = v γ = u + w, γ = w + v + u, (4) u = u(,, ), v = v(,, ) and w = w(,, ) are displacements in the directions of, and aes. Furthermore, ε, ε and ε are normal strains corresponding to normal stresses σ, σ and σ, whereas γ, γ and γ are shear strains corresponding to shear stresses τ, τ and τ. Generalised Hooke s law In linear elasticit, if the stress is sufficientl small, Hooke s law is used to represent the material behaviour and to relate the unknown stresses and strains. The general equation for Hooke s law is: ε ij = s ijkl σ kl = 3 k=1 3 l=1 s ijkl σ kl (5) where i, j = 1,, 3 In this case s ijkl is a fourth-rank tensor called elastic compliance of the material. ach subscript of s ijkl takes on the values from 1 to 3, giving a total of 3 4 = 81 independent components in s. However, due to the smmetr of both ε ij and σ kl, the elastic compliance s must satisf the relation: s ijkl = s jikl = s ijlk = s jilk (6) It follows from quation 5, therefore, that the generalised Hooke s law in quation 5 can be simplified to become: ε i = s ij σ j, where i, j = 1,.., 6 (7) This relationship reduces the number of s components to 36, as seen in the following linear relation between the pseudovector forms of the strain and stress: ε 1 s 11 s 1 s 13 s 14 s 15 s 16 σ 1 ε s 1 s s 3 s 4 s 5 s 6 σ ε 3 s = 31 s 3 s 33 s 34 s 35 s 36 σ 3 ε 4 s 41 s 4 s 43 s 44 s 45 s 46 σ 4 ε 5 s 51 s 5 s 53 s 54 s 55 s 56 σ 5 ε 6 s 61 s 6 s 63 s 64 s 65 s 66 σ 6 ε s 11 s 1 s 13 s 14 s 15 s 16 σ ε s 1 s s 3 s 4 s 5 s 6 σ ε s = 31 s 3 s 33 s 34 s 35 s 36 σ γ s 41 s 4 s 43 s 44 s 45 s 46 τ γ s 51 s 5 s 53 s 54 s 55 s 56 τ γ s 61 s 6 s 63 s 64 s 65 s 66 τ The s matri in this form is also smmetric. It therefore contains onl 1 independent (8) elements. The number of independent elements is obtained b counting elements in the upper right triangle of the matri, including the diagonal elements (i.e. 1 + + 3 + 4 + 5 + 6 = 1). Furthermore, if the material ehibits smmetr in its elastic response, the number of independent components in the s matri will be reduced even further. For eample, in the simplest case of isotropic materials whose elastic moduli are the same in all directions, onl three unique components (s 11, s 1, s 44 ) eist, as shown in quation 9. In the elastic range, these three coefficients of the ordinar isotropic model require three parameters (, v, G) for their definitions as follows: s 11 s 1 s 1 s 1 s 11 s 1 s s = 1 s 1 s 11 s 44 s 44 s 44 s 11 =, 1 v s 1 = s 44 =, 1, G G = (1 + v) (9) In cross-anisotropic materials, however, si unique components (s 11, s 1, s 13, s 33, s 44 ) and s 66 eist, as shown in quation 1. These si components of the ordinar cross-anisotropic model require five parameters ( v, h, v hv, υ hh, G hv ) for their definitions (Love 1944). s 11 s 1 s 13 s 1 s 11 s 13 s s = 13 s 13 s 33 s 44 s 44 s 66 (1) s 11 = 1 v, s 1 = hh v, s 13 = hv, h h v s 33 = 1 1, s 44 =, G hv = v, v G hv (1 + v hv ) s 66 = (s 11 s 1 ) Writing the stresses in terms of the strains would require quation 8 to be inverted, ielding: σ i = c ij ε j (11) c ij is a fourth-rank tensor called elastic stiffness of the material. The stiffness and compliance matrices in quation 11 are related in the following form: c = s 1 (1) Three components (c 11, c 1, c 44 ) eist for isotropic materials, whereas, in cross- anisotropic materials, five unique components (c 11, c 1, c 13, c 33, c 44, c 66 ) eist. It should be noted that a special crossanisotrop solution eists, where v = h and v vh = υ hh. quation 13 shows the si components of the elastic stiffness matri, whereas quations 14 19 define each of these si components: c 11 c 1 c 13 c 1 c 11 c 13 c c = 13 c 13 c 33 c 44 c 44 c 66 (13) c 11 = h ( v + h vhv ) (14) (1 + v hh )( v + v v hh + h vhv ) c 1 = h ( v v hh + h vhv ) (15) (1 + v hh )( v + v v hh + h vhv ) c 13 = h v v hv v + v v hh + h vhv v c 33 = ( 1 + v hh ) v + v v hh + h vhv c 44 = G = v (1 + v hv ) c 66 = v (1 + v hv ) = (c 11 c 1 ) Substituting quation 13 in 11 ields: σ σ σ τ τ τ c 11 c 1 c 13 c 1 c 11 c 13 c = 13 c 13 c 33 c 44 c 44 c 66 ε ε ε γ γ γ (16) (17) (18) (19) () Derivation of the solutions It is convenient to use the three-dimensional Cartesian coordinate sstem and make an assumption that displacement functions u(,, ), v(,, ) and w(,, ) in the, and directions, respectivel, ma be represented using double trigonometric functions, as detailed below. With this approach, the and dependencies of the displacement functions u(,, ), v(,, ) and w(,, ) are accommodated b means of analtical double integral Fourier transform, with the dependence approimated b using closed-form solutions. The Fourier transform of the displacements emplos transform pairs that are 3 Journal of the South African Institution of Civil ngineering Volume 59 Number 1 March 17
defined in terms of the Fourier parameters in the and directions (ξ and ξ ) as follows: ũ(ξ, ξ, ) = u() sin(ξ ) cos(ξ )dξ dξ (1) ṽ (ξ, ξ, ) = v() cos(ξ ) sin(ξ )dξ dξ () w (ξ, ξ, ) = w() cos(ξ ) cos(ξ )dξ dξ (3) ũ(ξ, ξ, ), ṽ (ξ, ξ, ) and w (ξ, ξ, ) are the Fourier transforms of displacement functions about the coordinate (Sneddon 1951). The procedure that follows is to epand quation and to epress strains in terms of elastic displacements based on quation 4. After that the Fourier transforms of the displacements using quations 1, and 3 are applied. Then the resulting functions are substituted into quations 1 3 to ield their Fourier transforms as follows: ũ(ξ, ξ, )(ξ (c 11 c 1 ) + c 11 ξ ) + ξ ξ (c 11 + c 1 ) ṽ (ξ, ξ, ) + ξ (c 13 + c 44 ) w (ξ, ξ, ) c ũ(ξ, ξ, ) 44 = (4) ξ, ξ (c 11 + c 1 )ũ(ξ, ξ, ) + ṽ (ξ, ξ, ) (ξ (c 11 c 1 ) c 11 ξ ) + ξ (c 13 + c 44 ) w (ξ, ξ, ) c ṽ (ξ, ξ, ) 44 = (5) ũ(ξ (c 13 + c 44 ) ξ, ξ, ) ṽ(ξ + ξ, ξ, ) + c w (ξ, ξ, ) 33 c 44 w (ξ, ξ, ) (ξ + ξ = (6) quations 4 6 can be simplified further b representing a differential function with respect to as λ = as well as making the following substitutions: a = c 11 c 1 c 44, b = c 11 + c 1 c 44, c = c 13 + c 44 c 44, d = c 44 c 33, f = c 13 + c 44 c 33 (7) and ξ = ξ + ξ (8) Using quations 7 and 8 it becomes convenient to eliminate ũ(ξ, ξ, ) and ṽ(ξ, ξ, ) from quations 4 6, then simplif to obtain: (λ 4 t 1 ξ λ + t ξ 4 )w = (9) The roots of quation 9 are determined as: λ = ±ξ t 1 ± t 1 4t (3) t 1 = (b + (a + d) c f ), t = a d + b d from quation 7. Putting back the differential function in quation 9 ields: 4 t 4 1ξ + t ξ4 w = (31) With reference to the roots in quation 3, the solution form of quation 31 depends on the sign of the coefficient t 1 4t. Furthermore, and this is ver important, depending on the numerical integration methods used, it ma be necessar to modif the solution for w to obtain stable and accurate responses of the pavement structure. The solutions presented hereunder were based on the numerical methods used in this research. 1. Solution 1: t 1 4t >, where h > v w = w (ξ, ξ, ) = C 1 (ξ)e λ 1 + C (ξ)e λ 1 + C 3 (ξ)e λ + C 4 (ξ)e λ (3) t λ 1 = ξ 1 + t1 4t t, λ = ξ 1 t1 4t, and C 1 (ξ), C (ξ), C 3 (ξ), and C 4 (ξ) are coefficients of integration determined, as described later, b using boundar loading conditions. In order to obtain stable and accurate results of pavement responses, quation 3 was modified to: w (ξ, ξ, ) = C 1 (ξ) cosh(r ) e r 1 + C (ξ) cosh(r ) e r 1 C 3 (ξ) sinh(r ) e r 1 C 4 (ξ) sinh(r ) e r 1 (33) r 1 = ξ r = ξ t 1 t1 4t + t 1 + t 1 4t, and t 1 + t1 4t t 1 t 1 4t. Solution : t 1 4t =, where h = v w (ξ, ξ, ) = e ξ C 1 (ξ) ξ C 3 (ξ) ξ ξ ξ 3c + C 5 (ξ) 11 c 1 ξ ξ (c 11 + c 1 ) ξ + e ξ C (ξ) ξ + C 4 (ξ) ξ ξ ξ 3c + C 6 (ξ) 11 c 1 + ξ (34) ξ (c 11 + c 1 ) ξ 3. Solution 3: t 1 4t <, where h < v w (ξ, ξ, ) = C 1 (ξ)e λ 1 + C (ξ)e λ 1 + C 3 (ξ)e λ + C 4 (ξ)e λ (35) λ 1 = ξ λ = ξ t 1 + t1 4t, and t 1 t1 4t For stable and accurate results of pavement responses, quation 35 was modified to: w (ξ, ξ, ) = C 1 (ξ) cos(r ) e r 1 + C (ξ) cos(r ) e r 1 C 3 (ξ) sin(r ) e r 1 C 4 (ξ) sin(r ) e r 1 (36) r 1 = ξ t 1 and r = ξ t 1 4t Solutions for the remaining Fourier transformed displacements Substituting quations 33, 34 and 36 into quations 4 and 5, the solutions for ũ(ξ, ξ, ) and ṽ(ξ, ξ, ) are derived. As an eample, solutions for ũ(ξ, ξ, ) and ṽ(ξ, ξ, ) for the case where h = v (solution ) are obtained as follows: ũ(ξ, ξ, ) = C (ξ)e ξ + C 4 (ξ)e ξ + C 5 (ξ)e ξ + C 6 (ξ)e ξ (37) ṽ(ξ, ξ, ) = eξ C 1 (ξ) + C 3 (ξ) ξ + C 5 (ξ) ξ ξ ξ + e ξ C (ξ) + C 4 (ξ) ξ ξ + C 6 (ξ) ξ (38) ξ Solutions for ũ(ξ, ξ, ), ṽ(ξ, ξ, ) and w (ξ, ξ, ) are then substituted into quations 4 and to determine the Fourier transforms of normal and shear Journal of the South African Institution of Civil ngineering Volume 59 Number 1 March 17 31
p (,, ) Applied load/stresses P/σ Contact stresses Stress rotation Surfacing: h 1, 1v, 1h, v 1v, v 1h Figure 3 Surface rectangular vertical load (p (,, ) = ) stresses σ (ξ, ξ, ), σ (ξ, ξ, ), σ (ξ, ξ, ), τ (ξ, ξ, ), τ (ξ, ξ, ), and τ (ξ, ξ, ). Boundar condition surface rectangular vertical loading For the loading case shown in Figure 3, there is onl a single uniforml distributed surface rectangular vertical load, P, whose sides are a and b in dimensions. Boundar conditions for rectangular loads acting on a surface of a semi-infinite medium shown in in Figure 3 ma be represented b taking into consideration the equilibrium between eternal and internal vertical and shear stresses, as shown below: When a and b then: σ 1 (ξ, ξ, = ) τ 1 (ξ, ξ, = ) p (ξ, ξ, = ) = τ 1 (ξ, ξ, = ) (39) p (ξ, ξ ) = a a b p cos(ξ ) cos(ξ ) d(ξ )d(ξ ) b π = p sin(ξ ) sin(ξ ) and ξ ξ π P p = ( a)( b) In addition, when > a and > b then: σ 1 (ξ, ξ, = ) τ 1 (ξ, ξ, = ) = τ 1 (ξ, ξ, = ) (4) Pavement responses B appling Fourier inverse transformation of all the Fourier transformed solutions, it is possible to determine solutions for pavement responses at an point in a pavement structure. u(,, ) = 1 π ũ(k, k, ) sin(ξ ) cos(ξ )dξ dξ (41) v(,, ) = 1 π ṽ(ξ, ξ, ) cos(ξ ) sin(ξ )dξ dξ (4) Table 1 Profile of the hpothetical pavement structure Laer tpe Pavement response Stresses and strains Figure 4 Hpothetical pavement structure Laer number -Modulus vert (MPa) -Modulus hori (MPa) P-Ratio vert Base laer: h, v, h, v v, v h Sub-base laer: h 3, 3v, 3h, v 3v, v 3h Selected laer: h 4, 4v, 4h, v 4v, v 4h Sub-grade: h 5, 5v, 5h, v 5v, v 5h P-Ratio hori Thickness (mm) AC 1 81 18 618.14.13 1 AC 19 781 19 767.15.15 1 AC 3 19 33 19 8.15.15 1 AC 4 18 846 15 583.1.11 1 G1 5 45 45.35.35 37.5 G1 6 45 45.35.35 37.5 G1 7 45 45.35.35 37.5 G1 8 45 45.35.35 37.5 C3 9 5 5.35.35 75 C3 1 5 5.35.35 75 C3 11 5 5.35.35 75 C3 1 5 5.35.35 75 G7 13 5 5.45.45 37.5 G7 14 5 5.45.45 37.5 G7 15 5 5.45.45 37.5 G7 16 5 5.45.45 37.5 SG 17 39 39.4.4 Semi-infinite Slip rate 3 Journal of the South African Institution of Civil ngineering Volume 59 Number 1 March 17
Stress, S (MPa) 4 6 Stress, S (MPa) 4 1 1 3 4 3 4 5 5 6 6 7 7 This stud S FM S This stud S FM S Figure 5 Stress distribution under the centre of the load with depth between this stud and proprietar FM package w(,, ) = 1 σ (,, ) = 1 σ (,, ) = 1 σ (,, ) = 1 τ (,, ) = 1 τ (,, ) = 1 τ (,, ) = 1 π w (ξ, ξ, ) cos(ξ ) sin(ξ )dξ dξ (43) π σ (ξ, ξ, ) cos(ξ ) cos(ξ )dξ dξ (44) π σ (ξ, ξ, ) cos(ξ ) cos(ξ )dξ dξ (45) π σ (ξ, ξ, ) cos(ξ ) cos(ξ )dξ dξ (46) π τ (ξ, ξ, ) sin(ξ ) cos(ξ )dξ dξ (47) π τ (ξ, ξ, ) cos(ξ ) sin(ξ )dξ dξ (48) π τ (ξ, ξ, ) sin(ξ ) sin(ξ )dξ dξ (49) worked eamples Solutions developed in this stud were used to compute responses at different positions within a pavement structure. The pavement structure for which the simulation results are presented here is shown in Figure 4. Based on historical tre loading on South African roads, a tre vertical load of 1.5 kn on a rectangular patch of 31 mm b 38 mm resulting in a 39 kpa contact stress was used in the analsis (De Beer 8). All laers, ecept the asphalt laer, were modelled with isotropic, linear-elastic properties. The asphalt laer was modelled with cross-anisotropic, linear elastic properties. Information on all the laers, including sub-laers, is shown in Table 1. The sub-laering of the upper four main laers into four laers each resulted in a total of 17 laers, as shown in Table 1. The vertical stiffness in the asphalt sublaers shows a marginal increase close to the surface that ma be attributed to binder ageing, but more probabl to a slight reduction in the temperature conditions. The top and bottom asphalt sub-laers show significant reduction in the effective horiontal stiffness resulting from cracking initiating from both the top and bottom of the main asphalt laer. Discussion of results Figures 5 and 6 show comparisons between the numerical method developed in this stud and a proprietar FM package for stresses (S) and (S) as well as strains () and () distributions with depth, -direction, under the centre of a rectangular surface load of magnitude 1.5 kn (39 kpa contact stress). From these results it is clear that the closed-form solutions developed in this stud have achieved a ver good level of accurac, as their results compare well with results from a proprietar FM package. It is also important to mention here that the software developed from this work has become the analsis engine for the new SAPDM (South African Pavement Design Method). What is also evident in the strains plots is that, as the FM increases the sie of the elements at points far from where the load is acting, the accurac is reduced a little bit and the results start moving awa from the results of this stud. However, all in all, the agreement of the two methods, i.e. approimation b FM and closedform solution developed in this paper, is ver good. CONCLUSIONs 1. A numerical tool for analsis of an elastic multilaer sstem under the action of a surface rectangular load, considering both cross-anisotropic and isotropic material properties, has successfull been developed.. The numerical tool developed in this stud is capable of performing analses for an unlimited number of points in an elastic multi-laered pavement sstem with an unlimited number of laers, and on the surface where an unlimited number of uniforml distributed rectangular loads act. Journal of the South African Institution of Civil ngineering Volume 59 Number 1 March 17 33
Strain, (MPa) Strain, (MPa) 1.-4.+ 1.-4.-4 3.-4 4.-4 5.-4 1.5-4 1.-4 5.-5.+ 5.-5 1.-4 1.5-4 1 1 3 4 3 4 5 5 6 6 7 7 This stud FM This stud FM Figure 6 Strain distribution under the centre of the load with depth between this stud (closed-form) and proprietar FM package 3. The results shown in this paper confirm the accurac and reliabilit of the closedform theoretical solutions developed. 4. A ver good match of the stress results was obtained between the numerical tool developed in this stud and a proprietar FM package. 5. Differences in the strain results at points far from where the load acts, were observed. The differences seem to be emanating from the FM, which is a proprietar package. In conventional FM, stress computations come after strain computations, and the trends should have been similar. The reason for the differences is not clear, but the results, as measured, are nevertheless reported here. 6. The numerical tools developed can be used to improve the design, evaluation and analsis of multi-laered road/runwa pavement sstems. acknowledgments This research stud was part of the revision of the SAPDM, a project sponsored b the South African National Roads Agenc SOC Ltd (SANRAL) and the Council for Scientific and Industrial Research (CSIR). The incentive funding for rated researchers received from the National Research Foundation (NRF) is also gratefull acknowledged, as well as the support from Dr Hechter These for setting up the numerical problem together with Prof Rudi Du Pree to run the proprietar FM analses for comparative purposes. References Borodachev, N M 1995. Three-dimensional problem of the theor of elasticit in strains. Strength of Materials, 7(5 6): 96. Borodachev, N M 1. Construction of eact solutions to three-dimensional elastic problems in stresses. International Applied Mechanics, 39(6): 438. Borodachev, N M & Astanin, V V 8. Solution of a three-dimensional problem of the elasticit theor in terms of displacements for an isotropic elastic laer. Strength of Materials, 4(3): 38 315. Bufler, H 1971. Theor of elasticit of a multilaered medium. Journal of lasticit, 1(): 15 143. De Beer, M 8. Stress-In-Motion (SIM) A new tool for road infrastructure protection? Proceedings, International Conference on Heav Vehicles, WIM Session 7, 19 Ma, Paris/Marne-la-Vallée. rnian, P 1989. Static response of a transversel isotropic and laered half-space to general surface loads. Phsics of the arth and Planetar Interiors, 54: 353 363. Filonenko-Borodich, M 1963. Theor of lasticit. Moscow: Peace Publishers. Love, A 1944. A Treatise on the Mathematical Theor of lasticit. New York: Dover Publications. Maina, J W, Kawana F & Matsui, K 17. Numerical modelling of fleible pavement incorporating crossanisotropic material properties Part I: Surface circular loading. Journal of South African Institution of Civil ngineering, 59(1): 7. Oawa, Y, Maina, J W & Matsui, K 9. Linear elastic analsis of pavement structure loaded over rectangular area. Paper presented at the 88th Transportation Research Board Annual Meeting, 11 15 Januar, Washington, DC. (CD-ROM.) Oawa, Y, Maina, J W & Matsui, K 1. Analsis of multilaered half space due to rectangular moving load. Paper presented at the 89th Transportation Research Board Annual Meeting, 1 14 Januar, Washington, DC. (CD-ROM.) Sneddon, I N 1951. Fourier Transforms. New York: McGraw-Hill. 34 Journal of the South African Institution of Civil ngineering Volume 59 Number 1 March 17